Controller selection and placement in compressible turbulent flows

Transcription

1 Center for Turbulence Research Proceedings of the Summer Program Controller selection and placement in compressible turbulent flows By D. J. Bodony AND M. Natarajan Deciding how and where to control a fluid system often relies heavily on intuition, experience, and what actuators are readily available. While useful for flows for which experience exists, adopting such an approach for the control of novel, complex fluid systems may lead to suboptimal or ineffective control strategies. In this report we develop a procedure to use the structural sensitivity of a baseflow to estimate where an effective linear feedback controller might be placed without needing a priori knowledge. When applied to a simple high-subsonic diffuser, the method identifies candidate controller locations and types and demonstrates that the preferred locations depend on the type of controller and on the sensing variable.. Introduction Active and passive flow control is a now well-established means for affecting the performance of fluidic systems. Several examples exist for drag reduction (Lee et al. 998), separation reduction (Kale et al. 22), and aerodynamic noise reduction (Wei & Freund 26; Thomas et al. 28), to name a few. Several different actuator types are available that, together, impart disturbances over a wide range of length and time scales (Cattafesta III & Sheplak 2). Furthermore, a range of control approaches are possible depending on the level of fidelity required, e.g., reduced-order or full model fidelity (Rowley 25; Ahuja & Rowley 2; Colburn et al. 2), and whether optimal or robust control is desired (Bewley et al. 2; Kim & Bewley 27). It is common practice to decide what actuator type is to be used based on what is readily available. It is likewise conventional to guess, based on experience and data available in the literature, the likely best actuator location and forcing schedule. In jet noise reduction, for example, the nozzle lip is a commonly selected location based on receptivity arguments (Samimy et al. 27) and it is generally believed that wallnormal forcing is most effective. In boundary layers, active forcing near the most unstable frequency near the most receptive location has been found beneficial (Marquet et al. 29). Whether the chosen actuator type and location are optimal for a particular flow, however, have been considered in less detail, although there are several examples of adjoint-based control determining the optimal response for an assumed actuator type and location, as in Bewley et al. (2). Chen & Rowley (2), using the linearized complex-valued Gizburg-Landau equation, developed a method based on algebraic Ricatti and Lyapunov equations to optimally determine the location of actuator-sensor pairs in a one-dimensional, time-dependent setting. Although novel, their methods may not be computationally feasible for large, three-dimensional flows of compressible, vis- Department of Aerospace Engineering, University of Illinois at Urbana-Champaign

2 36 Bodony & Natarajan cous fluids. The type of actuator was not considered explicitly because of the simplicity of the dynamical system. We thus seek an alternative approach that is (a) generalizable to complex, threedimensional flow fields and (b) computationally feasible. As a trade-off we relax the optimally requirement and seek instead estimates of good actuator types and locations. Whether they are optimal requires an interative procedure whereas ours is a one-shot computation. In Section 2 the method is developed prior to application in Section 3. The report closes with conclusions given in Section Method The underlying concept of this analysis is that of the wavemaker, as used by Giannetti & Luchini (27) and included in the review of Chomaz (25), which is a region that selects the frequency of response in an absolutely unstable region. (Modifications must be made if the flow is absolutely stable but contains convective instabilities, but we do not consider this case here.) The wavemaker highlights the areas of a flow in which a small force of the right type will lead to the strongest response and can be found as follows. If the unactuated compressible Navier-Stokes equations can be written as q + N( q) =, (2.) t then the linearized equations for q, the change to the base flow q e due to actuation f( x,t), with corresponding adjoint equation, are q t + A[ q e] q = f, q t + A [ q e ] q =. (2.2) The structural sensitivity is defined as the change in the eigenvalues of A[ q e ],which amounts to assuming an exponential time dependence of q ( x,t) = ˆ q( x)exp{σt} and similar for q, when q e is changed to q e + q. For flow control, we want to change (and possibly stabilize) an unstable mode with eigenvalue σ, and the amount of change of the eigenvalue can be shown to be ˆ q ˆ f dv δσ = ˆ q ˆ q dv, (2.3) where the integrals are taken over space and the forcing f has been written ˆ f exp{σt}. Because both the forward q and adjoint q fields are used we are ensured that the flow will be receptive and amplify the disturbances, thus increasing control effectiveness. Note that the adjoint field is normalized according to ˆ q ˆ q dv =. The real power of the method is that a simple optimization of ˆ f yields the preferred actuator type and location: ˆ f opt = arg max δσ. (2.4) ˆ f The resulting actuator ˆ fopt is guaranteed, in a linear context, to most effectively alter the base state of interest. The problem can be made simpler by assuming a linear feedback approach so that the actuator depends on the flow ˆ f = C( x)ˆ q. (2.5)

3 The wavemaker is then defined as Controller selection and placement 37 λ( x) = ˆ q ˆ q, (2.6) such that the eigenvalue change δσ can be bounded δσ C λ( x). In two-dimensions we assume that the matrix C( x) can be written c c 4 C( x) = exp{ (x x ) 2 /l 2 x (y y ) 2 /l 2 y}, (2.7) c 4 c 44 which localizes the control about the point x = (x,y ) with length scale l = (l x,l y ). (In what follows all of the c ij, x, y, l x and l y are determined through optimization.) Observe that the c ij determine the relationship between the forcing type and the sensing quantity. 2.. Computational method The primary computational cost is in computing the eigenvectors q and q, of A[ q e ] and A [ q e ], respectively, that correspond to the eigenvalue of interest. We discretize the linearized forward equation, Eq. (2.2), using high-order summation-by-parts operators (Strand 994) with boundary conditions implemented using the simultaneousapproximation-term methodology (Carpenter et al. 994; Svärd et al. 27; Svärd & Nordström 28). The Cartesian coordinates x are mapped to non-orthogonal curvilinear coordinates ξ through the smooth and one-to-one mapping x = X( ξ). The viscous terms are written in expanded form to improve eigenvalue/eigenvector convergence. The discrete version of A( q e ) is created in sparse matrix form using PETSc (Balay et al. 22b,a, 997) and its eigenvalues and eigenvectors are computed using the shift-andinvert method of SLEPc (Hernandez et al. 25) with the MUMPS (Amestoy 22) LUfactorization used for preconditioning the Krylov-Schur iterations. The discrete adjoint is formed by taking the complex conjugate transpose of the discrete version of A( q e ). Determining the equilibrium solution q e ( x) to be a true steady-state solution and not the time average of an unsteady solution is important; we used the selective frequency damping method of Åkervik et al. (26). The non-linear equations are modified to include a damping term q + N( q) = χ( q w), t where the target state w is determined by the simple ODE w = q w, where χ and are parameters related to the unstable eigenvalues of A. The impact of finding the equilibrium solution q e rather than a time average can be seen in Figure for the mean velocity field for the problem described in the next section. 3. Results Our target application is loosely based on the boundary layer control for S-duct inlets where a high-subsonic flow is sharply turned and slowed down before entering an engine inlet. The specific geometry chosen for this study is a sharp-angled 5 degree diffuser with a straight upper wall (with slip conditions) and bent lower no-slip wall. The inlet

4 38 Bodony & Natarajan Figure. Mean x-velocity and streamlines for the Mach.7 subsonic diffuser. Left image: time average. Right image: equilibrium solution Figure 2. Partial forward eigenvalue spectrum (left) and the corresponding unstable eigenvector (right). The unstable mode is encircled in the left figure. channel height is h and the inlet conditions correspond to a Mach.7 base flow with a boundary layer of thickness.h. The Reynolds number Re, defined as ρ c h/µ, is 2,. Using the equilibrium base flow in Figure a portion of the forward eigenvalue spectrum is shown in Figure 2 where one globally unstable eigenmode is found with an eigenvalue of σh/c = i. The mode, which exists in both the upstream boundary layer and in the separation bubble, is the one that is to be stabilized. The question is where to place the actuator, and what type of actuator to choose, to most effectively manipulate the base flow to modify this mode. The forward solution indicates that the mode has greatest amplitude within the separation bubble. The adjoint eigenvalue spectrum is shown in Figure 3 along with the adjoint eigenvector corresponding to the globally unstable forward mode. The adjoint solution indicates the

5 Controller selection and placement Figure 3. Partial adjoint eigenvalue spectrum (left) and the corresponding unstable adjoint eigenvector (right). The unstable mode is encircled in the left figure Figure 4. Contour plot of the wavemaker showing the structural sensitivity of the globally unstable mode. receptivity is greatest near to, and upstream of, the corner of the diffuser. There is very little receptivity within the separation bubble itself. As indicated in Section 2 the most opportune regions of a flow to place localized feedback control are located where both the amplification and the receptivity are large. The wavemaker λ( x) bounds this region over all possible actuator pairs. Figure 4 shows that the corner, as expected, is an important region in controlling the unstable mode, as expected. Note that there is an extended region upstream of the corner that shows enhanced controllability and a limited, but not absent, region downstream of the corner. 3.. Controller selection and location The wavemaker in Figure 4 gives an overall view of the flow s structural sensitivity but is too coarse of a measure for use in selecting a specific localized feedback control strategy. We thus reprocess the forward and adjoint fields using the optimization condition in Eq. (2.4) and the linear feedback assumption in Eq. (2.7) to evaluate where specific actuators should be placed and what variables should be used to sense. For a mass source, where f = [f,,,] T exp{σt}, the results are shown in Figure 5(left) where the actuator separation based on which variable is sensed is apparent. All actuators are to be optimally located at or upstream of the corner. When either the density, streamwise momentum, and total energy (e.g., the pressure) is sensed, the actuator is to be placed

6 4 Bodony & Natarajan.5 mass source streamwise momentum source Figure 5. Optimal actuator locations for mass source (left) and streamwise momentum source (right) depending on the variable used for sensing..5 flow-normal momentum source total energy source Figure 6. Optimal actuator locations for vertical momentum source (left) and total energy source (right) depending on the variable used for sensing. several boundary layer thicknesses upstream of the corner, while a vertical velocity sensordriven actuator is better located at the corner. If, instead, a streamwise momentum source is chosen the results of Figure 5(right) suggest a very different actuator location strategy. Again a vertical velocity sensor is to be located near the corner but now all of the other sensing variables yield a most effective actuator that should be placed right at the corner. A survey of vertical momentum and total energy sources is given in Figure 6 where more actuator clustering near the corner is observed. Taken together Figure 5 and Figure 6 suggest that (a) there is important dependence on the actuator type and location on the sensing variable and (b) the newly developed method appears to build upon experience of how to control separation in this scenario. One additional, and useful, outcome of this approach is that the effectiveness of each actuator/sensor pair is quantified so that comparisons between different pairs can be made. The gains, defined as δσ, are tabulated in Table and show that mass and streamwise momentum sources with total energy sensing yield the most dominant effectiveness. Some pairs are seen to be ineffective and therefore avoided based on the present criteria.

7 Controller selection and placement 4 Sensor Actuator ρ ρu ρv ρe ρ ρu ρv ρe Table. Control effectiveness gains for the actuator/sensor pairs of Figure 5 and Figure Conclusions An actuator/sensor selection tool using the concept of the wavemaker has been developed, implemented and tested on a two-dimensional high-subsonic diffuser. The concept of guiding the selection of actuators based on linear theories appears to be useful and has shown that the best placement depends on what particular actuator/sensor variable pairs are selected. Quantitative ordering of the control effectiveness is a natural outcome of the process and can be used to rank actuator strategies. The optimality of the estimates remains to be tested. Acknowledgments The authors gratefully acknowledge the financial support of Rolls-Royce, North America, on portions of this work. Dr. Jack (Jagdish) S. Sokhey is the program manager. Computer support was provided by the Stanford Center for Turbulence Research and by the National Science Foundation (XSEDE contract TG-CTS94). DJB is extremely grateful to Dr. Joe Nichols (CTR) and Prof. Peter Schmid (LadHyX, Ecole Polytechnique, Paris) for discussions of this work. REFERENCES Ahuja, S. & Rowley, C. W. 2 Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. J. Fluid Mech. 645, Åkervik, E., Brandt, L., Henningson, D. S., Hoepffner, J., Marxen, O. & Schlatter, P. 26 Steady solutions of the Navier-Stokes equations by selective frequency damping. Phys. Fluids 8 (682), 4. Amestoy, P. 22 MUMPS: a multifrontal massively parallel sparse direct solver. http: //mumps.enseeiht.fr, last accessed October 22. Balay, S., Brown, J.,, Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F. & Zhang, H. 22a PETSc users manual. Tech. Rep. ANL-95/ - Revision 3.3. Argonne National Laboratory. Balay, S., Brown, J., Buschelman, K., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F. & Zhang, H. 22b PETSc Web page. Balay, S., Gropp, W. D., McInnes, L. C. & Smith, B. F. 997 Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools in Scientific Computing (ed. E. Arge, A. M. Bruaset & H. P. Langtangen), pp Birkhäuser Press.

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